eath's  Mathematical  Monographs 

|\  Issued  under  the  general  editorship  of 

Webster  Wells,  S.  B. 

f?rofc88or  of  Mathematics  in  the   Massachusetts   Institute  of  Technology 


ON   TEACHING 


GEOMETRY 


BV 


FLORENCE    MILNER 

L)ei«oii    Univkmity  School 

Dk'iHOIT,    MlCHlOAN 


QPi^Ql 


Heath  &  Co.,  Publishers 

^  C^O  New  York  Chicago 


Price,   Ten  Cents 


The  Ideal  Geometry 

Must  have 

Clear  and  concise  types  of  formal  dem- 
onstration. 

Many  safeguards  against  illogical  and 
inaccurate  proof. 

Numerous  carefully  graded,  original 
problems. 

Must 

Afford  ample  opportunity  for  origin- 
ality of  statement  and  phraseology 
without  permitting  inaccuracy. 

Call  into  play  the  inventive  powers 
without  opening  the  way  for  loose 
demonstration. 


Wells's  Essentials  of  Geometry 
meets    all   of  these    demands. 


Half  leather y  Plane  and  Solid,  J^Q  pages.  Price  $1.2^. 

Plane,  75  cents.      Solid,  75  cents. 


.  C.  H  EATH  &  CO.,  Publishers 


BOSTON 


NEW     YORK 


CHICAGO 


^^\^^  .^^STNUT  HILL,  Ma..;  ^ 

THE    SKIFULL    GRADATION 
OF    ORIGINAL    WORK 

IN    800    EXERCISES 

Is    A     Marked     Feature     of    Wells's 
Essentials    of    Geometry 


As  soon  as  the  art  of  rigorous  logic  is  acquired,  the  simpkr  and 
more  axiomatic  steps  are  omitted  from  the  given  proof,  and 
the  student  is  required  to  supply  them  for  himself.  Ample 
opportunity  for  originality  of  statement  is  afforded,  but  no  inaccuracy 
permitted.  The  inventive  powers  are  called  into  play  without  opening 
the  way  for  loose  demonstration.  As  power  of  independent  thought 
and  reasoning  grows,  the  student  is  given  fewer  helps,  and  finally,  is 
entirely  dependent  on  himself  to  make  his  constructions  and  prove  his 
propositions. 

sturdy  self-reliance, 
resourcefulness, 
and  ingenuity, 
are  the  results. 


I  commend  particularly  that 
feature  of  the  Geometry  in  which 
the  details  of  proof  are  left  grad- 
ually to  the  pupil. 

GEO.   BUCK. 
Steele  High  School,  Dayton,  O. 


I  like  the  original  exercises 
which  are  not  at  first  too  difficult, 
but  by  their  gradation  encourage 
and  stimulate. 

G.  K.   BARTHOLEMEW. 
English  School,  Cincinnati,  O. 


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ESSENTIALS  OF  ALGEBRA 

By  WEBSTER  WELLS,  S.B., 

Professor  of  Mathematics  in  the  Massachusetts  Institute  of  Technology. 


This  book  fully  meets  the  most  rigid  requirements  now  made 
in  secondary  schools.  Like  the  author's  other  Algebras,  it  has 
met  with  marked  success  and  is  in  extensive  use  in  schools  of 
the  highest  rank  in  all  parts  of  the  country. 

The  method  of  presenting  the  fundamental  topics  differs  at 
several  points  from  that  usually  followed.  It  is  simpler,  more 
logical  and  more  philosophical,  yet  by  reason  of  its  admirable 
grading  and  superior  clearness  The  Essentials  of  Algebra  is  not 
a  difficult  book. 

The  examples  and  problems  number  over  three  thousand  and 
are  very  carefully  graded.  They  are  especially  numerous  in  the 
important  chapters  on  Factoring,  Fractions,  and  Radicals.  All 
of  them  are  new,  not  one  being  a  duplicate  of  a  problem  in 
the  author's  Academic  Algebra. 

In  accurate  definitions,  clear  and  logical  demonstrations,  well 
selected  and  abundant  problems,  in  systematic  arrangement  and 
completeness,  this  Algebra  is  unequalled. 


Half  leather.    ^58  pages.     Introduction  price,  $1.10. 


D.  C.  HEATH  &  CO.,  Publishers,  Boston,  New  York,  Chicago 


FACTORING 

AS   PRESENTED   IN 

WELLS'  ESSENTIALS  ^/ALGEBRA 

I.    Advanced  processes  are  not  presented  too  early. 
II.     Large  amount  of  practice  work. 

No  other  algebra  has  so  complete  and  well-graded  development  of 
this  important  subject,  presenting  the  more  difficult  principles  at  those 
stages  of  the  student's  progress  when  his  past  work  has  fully  prepared 
him  for  their  perfect  comprehension. 

The  Chapter  on  Factoring  contains  these  simple  processes  : 

CASE    L      When  the  terms  of  the  expression  have  a  common  monomial  factor 

thirteen  examples, 
n.      When  the  expression  is  the  sum  of  two  binomials  which  have  a  common 
binomial  factor  —  twenty  examples. 
IIL      When  the  expression  is  a  perfect  trinomial  square — twenty-six  examples, 

IV.      When  the  expression  is  the  difference  of  two  perfect  square? fifty-five 

examples. 
V.      When  the  expression  is  a  trinomial  of  the  form  x^-\-  ax-\-b 
—  sixty-six  examples. 

VL      When  the  expression  is  the  sum  or  difference  of  two  perfect  cubes twenty 

examples. 
Vn.      When  the  expression  is  the  sum  or  difference  of  two  equal  odd  powers  of 
two  quantities  —  thirteen  examples. 

Ninety-three  Miscellaneous  and  Review  Examples. 
Further  practice  in  the  application  of  these  principles  is  given  in  the  two  following 
chapters — Highest  Common  Factor  and  Lowest  Common  Multiple. 

In  the  discussion  of  Quadratic  Equations,  Solution  of  Equations  by 
Factoring  is  made  a  special  feature. 

Equations  of  the  forms  Ar^—j';\r  — 24  =  0,  2x'^  —  x=-0,  x'^ -\- ^x^  —  x  —  a. 

=  0,  and  .AT?—  I  =0  are  discussed  and  illustrated  by  thirty  examples. 
The  factoring  of  trinomials  of  the  form  ^x~-^6x-\-c  and  ax'^-^l>x'^-{-c, 

which  involves  so  large  a  use  of  radicals,  is  reserved  until   Chapter 

XXV,  where  it  receives  full  and  lucid  treatment. 

The    treatment    of  factoring   is   but    one  of  the  many  features  ol 
superiority  in  Wells'  Essentials  of  Algebra. 

Half  Leather,  JjS  pages.      Frice  $1.10. 

D.     C.     HEATH     &     CO.,     Publishers 

BOSTON  RnOTl/^ll  WyORK  CHICAGO 

CHESTNUT  Hitr.  M/^t 


(i)  in  general  excellence 
(2)   in  special  fitness  for  use 

Central  High  School, 
Philadelphia,  Pa. 
I  have  examined,  with  interest  and  appreciation,  Wells's  Essen- 
tials of  Geometry  both  with  reference  to  its  improvements  over  his 
former  edition  and  also  with  reference  to  its  excellence  when  com- 
pared with  others  of  the  present  date.  I  have  no  hesitation  in 
saying  that  it  seems  to  me  superior  both  in  general  excel- 
lence and  special  fitness  for  use  in  the  schoolroom,  to  any  which 
I  have  seen.  George  W.  Schock, 

Prof  ess  07'  of  Mathematics. 


(i)   in  the  order  of  theorems 

(2)  in  the  proof  of   corroUaries 

(3)  in  the  grading  of  the  original  exercises 

(4)  in  the  opportunity  for  original  work 

Boys'  High  School, 
Brooklyn,  N.  Y. 

In  the  order  of  theorems,  the  proof  of  corroUaries,  the  grading  of 

the  original  exercises,  in  the  diagrams  for  the  exercises,  and  in  the 

opportunity  for  original  work,  Wells's  Essentials  of  Geometry  is 

notably  superior.  C.  A.  Hamilton, 

Instructor  in  Mathematics. 


Decidedly  the  Best 

Central  High  School, 
Cleveland,  Ohio. 
Wells's  Essentials  is  decidedly  the  best  geometry  for  general 

school  work  so  far  published. 

Walter  D.  Mapes, 

Instructor  in  Mathematics. 


ESSENTIALS  OF  GEOMETRY 

PLANE  AND  SOLID 

By   WEBSTER    WELLS,    S.B. 
Professor  of  Mathe77iatics  in  the  Massachusetts  Institute 
of  Technology 


In  this  new  work,  issued  in  1899,  the  ideal  of  modern  teach- 
ing of  Geometry  is  made  practical  by  a  method  which  neither 
discourages  the  pupil  nor  helps  him  to  his  hurt.  The  author 
recognizes  the  needs  of  the  beginner,  and  meets  them  in  such  a 
way  as  to  arouse  his  interest  and  enthusiasm.  The  college  re- 
quirements are  heeded,  both  in  letter  and  spirit,  without  sacrifice 
of  organic  unity. 

The  exercises  are  about  800  in  number,  and  are  carefully 
graded.  An  important  feature  consists  in  giving  figures  and  sug- 
gestions in  connection  with  the  exercises.  In  Books  I  and  VI,  a 
figure  is  given  for  nearly  every  non-numerical  exercise ;  in  the 
other  books  they  are  given  less  frequently.  It  is  believed  that  with 
this  aid  the  exercises  are  brought  within  the  capacity  of  the  aver- 
age pupil,  and  that  his  interest  in  the  solution  of  the  original 
exercises  will  be  stimulated. 

Every  definition,  demonstration  and  discussion  has  been  sub- 
jected to  rigorous  criticism  in  order  to  secure  clearness,  brevity 
and  absolute  accuracy.  It  is  believed  that  no  other  text  in 
Geometry  is  so  free  from  ambiguous  and  loosely  constructed 
statements. 

The  Appendix  contains  rigorous  proofs  of  the  limit-statements 
used  in  connection  with  the  demonstrations  of  Book  IX. 


Half  leather.     Pages,  viii  -\-jgi.     Introduction  price,  $1.2^. 
Plane  Geometry,  separately,  y^  cents.     Solid  Geojuetry,  separately,  75  cents. 


D.  C.  HEATH  &  CO.,  Publishers,  Boston,  New  York,  Chicago 


Good  Advice  from  Harvard 

GEOMETRY 

"  As  soon  as  the  pupil  has  begun  to  acquire  the  art  of  rigorous 
demonstration,  his  work  should  cease  to  be  merely  receptive,  he 
should  be  trained  to  devise  constructions  and  demonstrations  for 
himself,  and  this  training  should  be  carried  through  the  whole  of 
the  work  of  Plane  Geometry.  Teachers  are  advised,  in  their 
selection  of  a  text-book,  to  choose  one  having  a  clear  tendency  to 
call  out  the  pupil's  own  powers  of  thought,  prevent  the  formation 
of  mechanical  habits  of  study,  and  encourage  the  concentration  of 
mind  which  it  is  a  part  of  the  discipline  of  mathematical  study  to 
foster."  —  Extract  from  the  Harvat'd  University  Catalogue^  igoo, 
page  304. 

Wells's  Essentials  of  Geometry 

fully  meets  these  broad  requirements,  and    meets  them 
more  successfully  than  any  other  book. 


Half  Leather,  Plane  and  Solid,  ^1.25. 
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BOSTON  NEW   YORK  CHICAGO 


^Br 


ON     TEACHING 
GEOMETRY 


BY 


FLORENCE     MILNER 

DETROIT  UNIVERSITY  SCHOOL 
DETROIT,    MICHIGAN 


BOSTON  COLLEGE  LI.  UAUif 
CHESTNUT  HILL,  MA6S. 

MATH,  DEPT. 


BOSTON,   U.S.A. 

D.  C.  HEATH   &   CO.,  PUBLISHERS 

1900 


Copyright,   1900, 
By  D.  C.  Heath  &  Co. 


150013 


ON    TEACHING   GEOMETRY. 

Of  writing  many  geometries  there  is  no  end. 
With  any  of  them,  or  without  them  all,  the  good 
teacher  will  get  good  results;  with  the  best  of 
them,  the  poor  teacher  cannot  rise  above  medioc- 
rity. Under  both  conditions,  however,  there  is 
wisdom  in  a  careful  choice,  for  a  strong  book  not 
only  lessens  the  labors  of  a  good  teacher,  buf 
makes  it  possible  for  a  class  to  get  some  value  out 
of  the  work  in  spite  of  poor  teaching.  Yet  we  as 
teachers  are  inclined  to  ask  too  much  of  text- 
books, and  we  expect  them  not  only  to  do  their 
own  work,  but  also  to  become  responsible  for  a 
large  share  of  ours.  It  is  the  province  of  the  text- 
book to  present  clearly,  according  to  its  established 
sequence,  the  subject  matter  of  geometry,  not  to 
teach  how  to  teach  it,  and  that  book  is  best  which 
has  in  it  least  of  anybody's  method,  even  of  thei 
present  writer's,  and  most  of  clearly  expressed] 
geometry.  So,  however  good  the  book,  there 
always  remains,  and  wisely  too,  much  for  the 
teacher  to  do. 

It  is  not  the  present  purpose  to  outline  any  par- 
ticular method  of  teaching,  but  to  call  attention 
to  a  few  general  lines  which,  in  harmony  with  the 


2  On  Teaching  Geometry. 

subject,  should  be  followed  out,  and  to  touch  upon 
some  points  in  a  wider  training  which  a  correct 
handling  of  the  subject  should  give. 

Geometry  differs  markedly  from  the  preceding 
studies,  and  success  or  failure  depends  largely 
upon  getting  at  the  beginning  the  right  point  of 
view.  To  play  a  game  of  chess  one  must  learn 
the  moves  of  individual  pieces.  These,  in  infinite 
combination,  but  always  under  the  same  fixed  rules, 
make  up  the  most  intricate  of  intellectual  games. 
Geometry  is  analogous.  We  expect  no  one  to  play 
a  game  of  chess  until  he  learns  the  moves;  we 
should  not  expect  a  pupil  to  work  intelligently 
in  geometry  until  he  is  helped  to  the  mental 
attitude  demanded  by  the  subject,  and  knows 
something  of  the  few  simple  truths  that  are  the 
guiding  thread  through  the  seemingly  intricate 
labyrinth. 

In  geometry  the  pupil  encounters  for  the  first 
time  formal  logic,  and  we  are  too  prone  to  plunge 
a  class  into  it  without  adequate  preparation.  Be- 
fore opening  the  book  at  all,  the  pupil  should  be 
I  taught  something  of  the  nature  of  logic,  something 
\  of  its  requisites,  something  of  its  method  of  work. 
Certain  texts  adhere  to  set,  formal  demonstrations, 
others  give  nothing  but  original  work,  while  be- 
tween these  two  extremes  are  all  grades  of  com- 
promise, with  more  or  less  practical  application  of 
estabhshed  truths.  Whether  we  are  working  out 
original  theorems  or  are  following  the  demonstra- 


On  Teaching  Geonietry.  3 

tion  of  a  conventional  proposition  or  are  using  our 
knowledge  in  the  solution  of  a  practical  problem, 
the  process  is  identical. 

Axioms  and  definitions  are  the  foundation  upon 
which  the  whole  superstructure  of  geometry  is 
builded,  and  in  the  beginning  a  class  should  be 
thoroughly  grounded  in  an  understanding  of  their 
nature  and  province.  Definitions  should  receive 
first  attention. 

If  you  and  I  are  to  carry  on  a  discussion  about 
an  apple,  we  must  agree  as  to  its  characteristics. 
If  you  define  it  as  "  the  fleshy  pome  or  fruit  of  a 
rosaceous  tree,"  while  I  insist  that  it  is  a  hard, 
black  mineral,  we  shall  never  get  far  in  our  argu- 
ment. If  you  say  that  "  a  triangle  is  a  plane  fig- 
ure bounded  by  three  right  lines,"  while  I  call  a 
solid  bounded  by  four  curved  surfaces  a  triangle, 
our  discussion  will  again  come  to  grief.  It  be- 
comes necessary,  then,  that  you  and  I  agree  upon 
a  common  conception  of  the  object  under  consider- 
ation, and  that  we  shall  so  describe  it  that  ambi- 
guity is  impossible.  This  is  the  province  of  the 
definition.  A  definition  is  such  a  description  as 
will  bring  up  to  all  minds  the  same  conception. 
The  definitions  in  geometry  have  long  been  agreed 
upon  by  mathematicians,  and  the  importance  of 
knowing,  and  knowing  with  verbal  accuracy,  the 
universally  accepted  description  of  geometric  con- 
cepts cannot  be  too  strenuously  insisted  upon,  and 
the  teacher  who  in  place  of  accurate  technical  Ian- 


4  On  Teaching  Geometry. 

guage  accepts  careless  and  verbose  statements, 
does  a  class  a  great  wrong.  A  lover  of  literature 
will  have  little  patience  with  one  who,  pretending 
to  give  Hamlet's  soliloquy  or  Milton's  sonnet  On 
his  Blmd?tess  would  dare  substitute  his  own  meagre 
words  for  the  matchless  language  of  the  great 
masters.  In  certain  directions  the  phraseology  of 
mathematics  has  crystallized  equally  with  other 
literature,  and  the  giving  of  familiar  definitions  is 
not  the  place  for  original  work,  —  that  comes  later. 
The  attitude  toward  axioms  should  next  be  made 
right.  Every  sane  mind  finds  itself  unconsciously 
in  possession  of  certain  knowledge.  We  know 
that  certain  things  are  true.  No  one  can  remem- 
ber when  he  learned  that  the  whole  of  anything  is 
greater  than  any  one  of  its  parts,  but  the  babe 
creeping  along  the  floor  has  a  practical  knowledge 
of  the  truth,  and  he  will  contest  it  with  you  to  the 
limit  of  his  physical  strength.  Later  he  will  for- 
mulate the  idea,  and  possibly  when  he  gets  into  the 
high  school,  some  teacher  will  set  him  to  learning 
this  axiom  as  though  it  were  something  new.  On 
the  contrary,  the  youth  should  be  taught  that  cer- 
tain facts  are  so  evidently  true  that  everybody 
must  needs  accept  them  or  be  counted  of  unsound 
mind.  When  he  begins  the  study  of  geometry, 
many  of  these  truths  should  be  put  into  compact 
form  for  future  use.  Here  again  comes  the  value 
of  systematic  and  even  stereotyped  phraseology. 
He  should  learn  now  to  state  in  the  technical  Ian- 


On  Teaching  Geometry.  5 

guage  of  mathematics  what  he  and  everybody  else 
have  long  known. 

When  the  pupil  has  been  put  in  the  right  atti- 
tude toward  axioms  and  definitions,  he  is  ready  to 
open  his  geometry  and  learn  how  to  use  them. 
Certain  general  definitions  should  be  studied.  He 
should  know  what  a  proposition  is ;  that  there  are 
various  kinds,  differing  in  purpose  and  in  form 
of  expression.  As  he  comes  to  them,  he  should 
discriminate  carefully  between  theorem,  problem, 
corollary,  and  lemma. 

Most  geometries  give  some  preHminary  work 
consisting  largely  of  definitions  and  other  discus- 
sion of  geometrical  concepts.  This  should  be 
studied  with  more  or  less  thoroughness,  as  the 
preparation  of  the  class  demands. 

Entering  upon  the  work  peculiar  to  geometry, 
we  come  to  the  opening  section,  which  may  treat 
of  perpendicular  straight  Hnes,  of  triangles,  or  of 
whatever  the  sequence  of  the  particular  text  in  use 
demands.  After  reading  the  first  theorem,  the 
hypothesis  should  be  isolated  that  the  pupil  may 
know  exactly  what  is  to  be  his  by  gift  of  this  same 
hypothesis,  and  he  should  be  taught  to  enumerate 
these  gifts.  Perhaps  the  enumeration  will  run  like 
this  :  "  A  Une,  a  point  without  that  line,  and  a  per- 
pendicular from  the  given  point  to  the  given  line." 
Show  the  class  that  it  is  not  sufficient  to  say  that 
a  line  and  a  point  are  given;  this  allows  one  to 
locate  the  point  within  the  line.     Whenever  they 


6  On  Teaching  Geometry. 

fail  to  be  thus  accurate  in  statement,  draw  figures, 
taking  such  latitude  as  they  leave  you  by  inaccu- 
rate statement.  A  few  illustrations  will  show  them 
the  necessity  of  talking  sharply  to  the  facts,  and 
the  pupils  will  soon  learn  to  hold  each  other  to 
statements  that  admit  of  no  ambiguity. 

A  certain  power  of  reconstructive  imagination 
should  also  be  cultivated.  Besides  stating  accu- 
rately the  conception  that  is  in  one's  own  mind, 
there  must  be  the  ability  to  construct,  rapidly  and 
clearly,  mental  pictures  of  all  statements  made  in 
class.  Every  sentence  uttered  should  add  some- 
thing to  the  picture  or  present  some  new  phase  of 
it.  Every  recitation  should  be  a  series  of  con- 
stantly changing  views  in  which  everybody,  in  his 
mind's  eye,  sees  the  same  things.  This  result  is 
not  easy  to  obtain,  but  it  is  possible,  and  the  degree 
of  excellence  reached  by  a  class  becomes  a  crucial 
test  of  the  teacher. 

Impress  upon  a  class  that  the  formal  statement 
of  a  proposition  is  always  a  genei'al  truth,  true 
of  all  figures  falling  under  the  given  conditions. 
These  statements  are  conventional  and  should  be 
as  accurately  stated  as  definitions  and  axioms. 

In  proving  a  proposition,  our  human  limitations 
require  us  to  fix  our  minds  upon  a  particular  case, 
and  accordingly  a  special  statement  follows. 

Given   i.    Line  AB. 

2.  Point  P  without  the  line. 

3.  Perpendicular  PD  upon  the  line 


On  Teaching  Geometry.  7 

To  prove,  that  PD  is  the  shortest  distance  from 
P  to  the  Hne  AB. 

If  there  is  more  than  one  conclusion,  bring  out 
the  facts  in  clear,  mathematical  language.  Do  not 
be  satisfied  to  let  a  class  say,  "  To  prove  that  from 
the  point  P  one  perpendicular  and  only  one  can  be 
let  fall  to  the  line  ABr  It  should  be  stated,  "  To 
prove :  first,  that  one  perpendicular  can  be  let  fall 
upon  the  line  AB\  and  second,  that  only  one  can 
be  let  fall  from  the  same  point  upon  the  line." 

This  may  seem  to  some  like  splitting  hairs,  but 
mathematics  is  an  exact  science,  and  in  the  begin- 
ning too  much  cannot  be  done  in  training  toward 
accurate  thought  and  exact  expression. 

Now  comes  important  work  in  preparing  for  a 
formal  demonstration.  Here  the  class  must  learn 
that  the  regular,  logical  form  of  every  argument  is 
a  syllogism.  Pupils  will  not  be  frightened  at  the 
new  word,  and  they  will  like  to  use  it  when  they 
comprehend  its  meaning.  Discuss  it  with  them. 
Explain  the  major  premise,  the  minor  premise, 
and  show  how  the  conclusion  inevitably  follows. 
Give  them  an  example  and  set  them  to  hunting  for 
others.    You  will  wonder  where  they  find  so  many. 

Next  teach  them  that  every  step  in  a  well-con- 
structed demonstration  is  taken  by  means  of  a  syl- 
logism. As  the  major  premise  must  always  enun- 
ciate a  general  truth,  the  major  premise  in  their 
first  demonstration  must  be  either  an  axiom  or  a 
definition,  for  these  are  the  only  general  geometric 


8  On  Teaching  Geometry.    . 

truths  in  their  possession.  The  minor  premise 
isolates  and  states  formally  the  special  case  now 
present  and  in  harmony  with  the  major  premise. 
The  conclusion  is  the  new  knowledge  revealed  by 
bringing  these  two  premises  together.  This  con- 
clusion in  turn  may  become  the  minor  premise, 
and  so  on  to  the  end  of  the  argument. 

Show  a  class  how  a  syllogism  may  be  worked 
out  from  some  of  the  early  definitions  and  axioms, 
letting  the  minor  premise  be  a  fact  given.  Tell 
them,  for  instance,  to  draw  the  line  CD  to  the 
point  D  in  the  line  AB,  making  the  angle  CDB  a 
right  angle.  This  is  the  fact  given,  and  if  dem- 
onstration were  to  follow,  might  constitute  the 
hypothesis.  Now  if  the  fact  is  of  any  value  in 
constructing  a  syllogism  it  must  fall  under  some 
general  definition  or  axiom.  Some  one  will  dis- 
cover that  the  definition  for  a  perpendicular  fits 
the  case.  Then  you  can  show  how  the  inevitable 
conclusion  that  CD  is  perpendicular  to  AB  must 
follow.  Finally  show  the  formal  structure  of  a 
syllogism  and  write  out  the  one  developed. 

1.  The  line  CD  meets  the  line  AB,  making  the 
angle  CDB  a  right  angle. 

2.  A  perpendicular  to  a  line  is  a  line  which 
makes  right  angles  with  a  given  line. 

3.  CD  is  perpendicular  to  AB. 

Repeat  the  process  until  you  make  them  see 
that  when  the  first  two  terms  are  rightly  selected 
and   agreed  upon,  there   is   no  escape   from  the 


On  Teaching  Geometry.  9 

conclusion.  It  follows  like  a  decree  of  inexorable 
fate. 

Every  demonstration  is  a  chain  of  syllogisms,  in 
all  of  which  the  major  premise  must  be  some  gen- 
eral conclusion  already  established,  either  axiom, 
corollary,  theorem,  definition,  or  algebraic  truth. 
Of  course  as  the  work  goes  on,  the  formal  state- 
ment that  involves  frequent  repetition  can  be  con- 
tracted, but  the  habit  of  looking  at  every  demon- 
stration from  this  side  is  invaluable. 

Text-books  do  not  follow  this  form,  for  they 
rarely  give  complete  demonstrations.  The  text  is 
an  outline  of  the  subject,  a  note-book,  keeping  the 
general  trend  of  the  argument  but  leaving  the 
pupil  to  fill  in  the  suggested  discussion.  Teachers 
rarely  insist  upon  so  closely  logical  a  demonstra- 
tion as  is  here  outlined,  but  experience  and  careful 
comparison  have  convinced  me  that  the  syllogistic 
method  closely  adhered  to  in  the  beginning  pro- 
duces results  that  come  from  no  other  kind  of 
work. 

In  the  illustration  just  used,  a  pupil  might  say 
and  with  truth,  "  CD  is  perpendicular  to  AB 
because  CDB  is  a  right  angle,"  and  "When  a  line 
meets,"  etc.,  but  that  is  making  a  statement  and 
then  going  back  to  try  to  make  the  statement 
good.  The  other  method  never  leaves  a  chance 
to  question  a  point,  for  one  follows  another  in  the 
logical  order  that  carries  conviction  at  every  step. 

If  you  announce  to  an  opponent  the  thing  of 


lo  On  Teaching  Geometry. 

which  you  expect  to  convince  him,  the  effect  is 
usually  to  arouse  his  antagonism,  and  he  will  then 
and  there  make  up  his  mind,  and  no  argument 
of  yours,  no  matter  how  convincing,  is  likely  to 
change  his  opinion.  He  may  listen  to  you  politely, 
but  when  you  are  through  he  will  probably  say, 
"Yes,  but  as  I  said  in  the  first  place."  Don't  give 
him  a  chance  to  say  anything  in  the  first  place, 
but  make  him  grant  your  premises,  one  after 
another,  then  when  your  conclusion  is  reached, 
there  is  nothing  left  for  him  to  do  but  accept  it. 

If  this  method  is  used  in  conventional  and 
formal  demonstration,  it  becomes  of  greatest  value 
in  original  work.  A  pupil  thus  trained  will,  when 
given  original  work,  have  a  definite  and  efficient 
method  of  attack.  He  will  first  study  his  hypoth- 
esis, isolating  each  individual  part  of  it.  Taking 
one  fact  for  a  minor  premise,  he  will  next  examine 
his  stock  in  trade  and  see  what  definition,  theorem, 
corollary,  or  other  authority  he  can  bring  to  bear 
on  the  case  in  hand.  He  will  draw  his  conclusion 
and  see  if  it  advances  his  argument  or  brings  him 
nearer  the  desired  end.  If  it  does  not,  then  he 
decides  that  his  minor  premise  is  wrong  and  re- 
turns to  his  hypothesis  for  another  fact,  and  con- 
structs the  syllogism.  Sooner  or  later  he  is  bound 
to  reach  the  right  conclusion. 

Sometimes  teachers  allow  pupils  to  say  that  such 
or  such  a  thing  is  true  "by  a  previous  proposi- 
tion."    This  is  a  much  abused  and  greatly  over- 


On  Teaching  Geometry.  ii 

worked  expression.  It  becomes  to  a  poor  student 
like  a  cloak  of  charity,  and  if  permitted,  will  be 
used  to  cover  a  multitude  of  hazy,  illogical,  and 
vague  ideas. 

Do  you  think  that  a  judge  in  court  would  admit 
as  evidence  the  statement  that  somewhere,  some- 
time, somebody  had  made  a  decision  in  a  case 
somewhat  similar  to  the  one  in  hand  ?  Indeed  not. 
The  lawyer  would  have  to  produce  his  authority, 
giving  title,  volume,  and  page  of  report,  with  date 
of  decision.  In  all  demonstrations  equal  accuracy 
should  be  demanded.  Not  that  chapter  and  verse 
be  given,  but  pupils  should  be  made  to  quote  geo- 
metric scripture  in  proof  of  every  conclusion.  And 
"quote"  here  means  exactly  what  the  word  indi- 
cates, not  a  slipshod  attempt  at  giving  the  idea. 

There  are  certain  definite  and  wide-reaching  re- 
sults that  should  come  from  the  right  teaching  of 
geometry.  Mention  has  already  been  made  of  the 
importance  of  accurate  and  exact  expression,  but 
this  must  be  preceded  by  equally  exact  and  accu- 
rate thinking.  Better  than  any  other  subject, 
geometry  will  train  the  youth  to  keep  close  watch 
and  ward  over  the  action  of  his  mind  and  accustom 
him  to  express  clearly  and  honestly  the  result  of 
his  own  mental  happenings.  The  ability  to  make 
wise  selection  under  varying  circumstances,  is 
repeatedly  demanded.  After  a  figure  is  drawn, 
the  pupil  should  examine  it  carefully.  He  may 
discover  in  it  many  things  which  he  knows,  from 


12  On  Teaching  Geometry. 

construction,  hypothesis,  or  other  conditions,  are 
all  true,  but  only  one  of  them  has  any  concern  with 
the  business  in  hand.  He  should  be  trained  to  see 
that  one  fact,  and  be  able  to  make  proper  use  of  it. 
Teach  him  to  go  straight  after  the  one  that  he 
needs,  and  make  it  serve  him. 

He  should  also  be  taught  that  certain  things  lie 
within  his  power  to  do,  while  others  are  as  abso- 
lutely beyond  his  control  as  are  the  turbulent 
waves,  the  floating  clouds,  or  the  sweeping  course 
of  the  planets. 

He  may  push  a  stone  from  the  edge  of  an  over- 
1  hanging  cliff,  but  after  that   he  must   let   it   go 
i   crashing  down  the  mountain  side.     He  may  elect 
I  to  let  fall  a  perpendicular  from  a  certain  point  to 
I  a  given  line,  but  he  cannot  dictate  where  it  shall 
strike  the  line.     He  may  know  certain  things  about 
I   two  figures,  and  it  may  be  wise  to  try  the  applica- 
f    tion  of  one  to  the  other.     He  may  lift  one  figure 
and  place  a  line  of  it  or  an  angle  upon  its  equal, 
:   but  there  his  control  over  the  matter  ceases.    From 
:   that  moment  he  is  under  law,  and  face  to  face  with 
\   the  eternal.     Let  him  stand  there  a  humble  spec- 
tator, knowing  that  till  heaven  and  earth  shall  pass, 
''one  jot  nor  one  tittle  shall  in  nowise  pass  from 
the  law  "  over  which  his  finite  mind  has  no  control. 
He  was  responsible  for  placing  the  figures  together, 
but  before  the  resulting  consequences  he  is  help- 
less.    He  may  watch  to  see  if  in  the  finality  there 
is  anything  that  concerns  him.    Here  again  he  will 


On  Teaching  Geometry.  13 

discover  various  things  accomplished ;  but  again 
he  must  recognize  the  one  conclusion  for  which  he 
has  been  striving,  must  isolate  it  from  the  rest,  and 
hold  it  up  to  view  with  the  strength  of  conviction. 

The  young  mind  is  naturally  unreasoning,  and 
often  utters  words  without  consciousness  of  a  defi- 
nite idea  back  of  them.     The  pupil  will  watch  the  ^ 
teacher  rather  than  himself,  to  determine  whether  I 
he  is  travelling  the  right  road.     It  is  very  easy  for  • 
a  teacher  unconsciously  to  take  this  responsibility,  j 
and  by  various  gentle  leading-strings  keep  the  pupil  '■■ 
in  the  path ;  but  such  work  is  not  teaching  geome- 
try.    If  a  class  leans  upon  you,  or  has  the  habit  of 
watching  you,  rest  assured  that  your  teaching  is  \ 
not  right.     If  the  development  is  correctly  carried 
on,  the  pupil  will  be  aroused  to  watch  his  own 
mind,  his  own  statements,  and  the  work  in  hand ; 
the  more  nearly  he  can  forget  j/oii,  the  better.     He  \ 
must  learn  to  discover  what  is  actually  happening 
in  his  own  mind  as  the  process  of  reasoning  goes  { 
on ;  he  must  be  trained  to  faith  in  his  own  con-  j 
victions,  and  to  fearlessness  in  expressing  them.  • 
Nothing  is  more  pitiful  than  a  mind  that  can  be 
shifted  from  any  position  by  an  incredulous  ques- 
tion.    At   first   your   pupils   will  not   think   inde- 
pendently.    Rouse  them  from  this  condition,  and  | 
force  them  to  a  conclusion  of  some  sort.     A  wrong  \ 
opinion  is  better  than  no  opinion  at  all,  for  activity  ^ 
is  better  than  stagnation. 

In  a  demonstration  the  teacher  must  be  con- 


14  On  Teaching  Geometry. 

stantly  in  the  attitude  of  the  doubting  Thomas, 
who  took  nothing  on  faith,  but  always  demanded 
evidence.  In  this  way  the  burden  of  proof  rests 
with  the  pupil,  and  his  aim  should  be  to  anticipate 
every  possible  question.  An  exceptionally  good 
teacher   of   geometry  says  that  the  right  results 

I  have  not  been  attained  until  a  pupil  not  only  ceases 
to  lean  upon  a  teacher,  but  is  also  able  to  stand 
up  against  a  teacher.  Do  not  rest  until  you  see 
your   pupils   thinking   independently,   and  at  the 

I  same  time  clearly,  fearlessly  talking  out  their  con- 

I  elusions. 

For  a  complete  setting  in  order  of  one's  mental 
house,  the  High  School  course  offers  nothing  better 
than  georaetry.  The  facts  learned  have  some  value, 
but  the  greatest  good  is  the  mental  poise,  the  clear- 

I  ness  of  vision,  and  the  honesty  of  expression  that 
it  develops.  In  addition  to  all  this,  a  pupil  rightly 
trained  will  learn  to  measure  his  own  strength,  will 
recognize  his  limitations,  and  will  bow  in  reverent 
respect  before  some  things  greater  than  himself. 


On  Teaching  Geometry.  15 

Note.  —  Some  teachers  may  be  interested  to  see  how 
the  preceding  discussion  would  tend  to  elaborate  in  reci- 
tation the  demonstrations  as  they  appear  in  our  best  text- 
books. It  has  seemed  wise,  accordingly,  to  add  such 
a  detailed  demonstration.  The  particular  theorem  is 
selected  because  it  offers  illustrations  of  more  points 
than  does  any  other  in  the  early  part  of  the  work. 

The  syllogistic  method  has  been  strictly  adhered  to 
until  the  last  few  steps  of  the  concluding  argument  are 
reached.  At  this  point  the  mind  works  so  rapidly  as  to 
become  impatient  of  talking  out  in  slow  words  what  it 
grasps  instantly,  if  the  preceding  argument  has  been 
properly  built  up.  You  have  seen  children  stand  domi- 
noes in  a  row  at  regular  intervals,  working  with  pains- 
taking care  to  get  them  rightly  placed.  A  single  touch 
upon  the  last  one  overthrows  it,  and  the  others  inevitably 
fall  in  rapid  succession.  So  in  this  argument.  When  the 
conclusion  that  CEC  cannot  be  a  straight  line  is  reached, 
all  the  rest  of  the  structure  comes  tumbling  down  so 
rapidly  that  no  time  is  left  to  do  more  than  to  watch  each 
domino  as  it  falls.  See  to  it  that  the  arguments  are 
rightly  placed  as  the  demonstration  is  built  up,  and  the 
rest  will  take  care  of  itself. 

Of  course  the  syllogisms  can  be  easily  repeated  in  this 
last  part  as  well,  and  it  is  often  a  good  exercise  to  allow 
a  class  to  supply  them. 


i6 


On  Teaching  Geometry. 


Book  I.     Proposition  VI.     Theorem.* 

From  a  given  point  witJioitt  a  straight  line  a  perpen- 
dicular can  be  drawn  to  the  line,  and  but  one. 

Given  the  line  AB  and  the  point  C  without  the  Hne. 

To  prove  (i)   that  from  the  point  C  one  J_  can  be 
drawn  to  the  Hne  AB ; 
(2)   that  only  one  J_  can  be  drawn  from  the 
point  C  to  the  line  AB. 

Proof.     I.    Draw  the  auxiliary  line  FG,  and  let  UK 
be  drawn  from  H 1^  to  AB. 
2.   At  a  given  point  in  a  straight  line  a  per- 
pendicular to  the  line  can  be  drawn,  and 
but  one.  (25) 

.-.  3.   BKisl.ioFG2XB. 
K 


F- 


H 


■a 


E\ 


D 


He' 

Apply  the  line  FG  to  the  line  AB,  and  move  it  along 
until  HX  passes  through  C.  Let  V  be  the  point  where 
H  falls.     Draw  the  hne  CB>. 

I .  CD  has  two  points,  C  and  D,  which  coincide  with 
points  in  HX  by  construction. 

*  From  Wells's  Essentials  of  Plane  and  Solid  Geometry. 


On  Teaching  Geometry.  17 

2.  But  one  straight  line  can  be  drawn  between  two 
points.  (Ax.  3.) 

.'.  3.  CD  coincides  with  HK,  and  CDB  coincides 
with  and  is  equal  to  KHG. 

1.  Xi7(S^  is  a  rt.  Z. 

2.  Z  CDB  =  AKHG. 
.'.  3.    CDB  is  a  rt.  Z. 
I.  Ci^^isart.  Z. 

2.  WTien  a  line  makes  a  right  angle  with  another  line, 
it  is  said  to  be  perpendicular  to  it.  (24) 

.-.  3.    CD  is  ±  to  AB. 

Hence  one  perpendicular  can  be  drawn  from  the  point 
C  to  the  Hne  AB. 

If  there  can  be  another  _L  from  C  to  AB,  let  it  be  C£. 
Produce  CD,  making  CD  =  CD.     Draw  C'£, 

1.  CD  =  CD  by  construction. 
£D  is  _L  to  CC  by  construction. 

2.  If  lines  be  drawn  to  the  extremities  of  a  straight 
line  from  any  point  in  the  perpendicular  erected  at  its 
middle  point,  they  make  equal  angles  with  the  perpen- 
dicular. (44) 

/.  3.  Z  C'£D  =  Z  C£D. 

1.  ACED  =  ACED, 

2.  Z  CED  is  a  rt.  Z  by  hypothesis. 
.-.  3.   Z  CED  is  also  a  rt.  Z. 
Add  Z  C^/)  and  Z  C'^Z>. 


1 8  On  Teaching  Geometry.  " 

1.  Z.  CED  is  a  rt  Z  by  hypothesis. 
Z  CED  is  a  rt.  Z  by  proof. 

2.  If  the  sum  of  two  adjacent  angles  is  equal  to  two 
right  angles,  their  exterior  sides  lie  in  the  same  straight 
line.  (37) 

.-.    3.    CEC  is  a  straight  line. 

1.  CEC  is  a  straight  line  by  proof. 
CDC  is  a  straight  line  by  construction. 

2.  But  one  straight  line  can  be  drawn  between  two 
points.  (Ax.  3.) 

.'.  3.  As  we  know  that  CDC  is  a  straight  line  by  con- 
struction, CEC  cannot  be  a  straight  hne. 

If  CEC  is  not  a  straight  line,  then  CED  +  CED  is 
not  =  to  two  rt.  A. 

If  CED  +  CED  is  not  equal  to  two  rt.  A,  then  CED^ 
which  is  half  of  this  sum,  is  not  a  rt.  Z. 

If  CED  is  not  a  rt.  Z,  then  CE  is  not  _L  to  AB. 

As  CjS"  is  any  other  possible  _L  from  C  to  AB,  then 
CZ^  is  the  only  _L  from  C  to  the  line  AB. 

Hence  there  can  be  only  one  perpendicular  from  C  to 
the  line  AB, 


THREE    (3)    POINTS 

FROM    MANY IN    WHICH 

WELL5'5   ESSENTIALS   OF 
GEOMETRY 

EXCELS : 

Accuracy 

No  other  Geometry  is  so  free  from  ambiguous  and  loosely 
constructed  statements.  Ever}-  definition  and  demonsti-a- 
tion  has  been  subjected  to  rigorous  criticism  in  order  to 
secure  clearness,  brevity  and  absolute  accuracy. 

Adaptation 

to  the  needs  of  beginners.  The  difiiculties  that  confront 
the  pupil  are  recognized  and  met  in  such  a  way  as  to  arouse 
his  interest  and  enthusiasm.  Propositions  and  original 
exercises  are  presented  in  a  manner  at  once  more  teachable 
and  more  educative  than  ever  before  attempted. 

Adequacy 

to  the  demands  of  the  colleges  and  technical  schools.  The 
entrance  requirements  are  heeded,  both  m  letter  and  spirit, 
without  sacrifice  of  organic  unity. 

{I.  Accurate  for  Everybody 
2.   Interesting  to  Pupils 
3.   Satisfactory  to  Teachers 

No  teacher   in  search  of  the  best  and  most  practical  text  on 
Geometry  can  afford  to  disregard  the  merits  of  Wells's  Essentials. 


D.  C.  HEATH  &  CO.,  Publishers 

BOSTON  NEW  YORK  CHICAGO 


EXERCISE  BOOK  IN  ALGEBRA 

^Designed  for  supplementary  or  review  work  in  connection  with 
any  text-hook  in  ^Igehra. 

By  MATTHEW  S.  McCURDY,  M.A., 

Itistructor  in  Mathematics  in  the  Phillips  Academy,  Andover,  Mass. 


This  book  is  designed  to  furnish  a  collection  of  exercises  similar 
in  character  to  those  in  the  ordinary  text-books,  of  medium  grade 
as  to  difficulty,  and  selected  with  special  reference  to  giving  an 
opportunity  for  drill  upon  those  subjects  which  experience  has 
shown  to  be  difficult  for  students  to  master. 

Though  intended  primarily  to  be  supplementary  to  some  regu- 
lar text-book,  a  number  of  definitions  and  a  few  rules  have  been 
added,  in  the  hope  that  it  may  also  be  found  useful  as  an  inde- 
pendent review  and  drill  book.     With  or  without  answers. 

Cloth.    Pages,  vi  -f-  220.     Introduction  price,  60  cents. 


ALGEBRA  LESSONS 

By  J.  H.  GILBERT. 

This  series  is  intended  for  supplementary  or  review  work,  and 
contains  three  numbers:  No.  i — To  Fractional  Equations, 
No.  2 — Through  Quadratic  Equations,  No.  3  — Higher  Algebra. 

Paper.     Tablet  form.    Price,  $1.44  per  do^en. 


REVIEW  AND  TEST  PROBLEMS 
IN  ALGEBRA 

By  S.  J.  PETERSON  and  L.  F.  BALDWIN. 

The  problems  in  this  manual  are  original  —  none  have  been 
copied  from  any  other  author.  They  illustrate  points  of  special 
importance,  and  are  sufficiently  varied  and  difficult  for  written 
drills  for  those  preparing  for  college  entrance  examinations. 

Paper.    8  j  pages.    Introduction  price,  ^o  cents. 

D.  C.  HEATH  &  CO.,  Publishers,  Boston,  New  York,  Chicago 


NEW   HIGHER  ALGEBRA 

By  WEBSTER  WELLS,  S.B. 

Professor  of  Mathe?fiatics  in  the  Massachusetts  Institute 

of  Technology 


The  first  358  pages  of  this  book  are  identical  with  the  author's 
Essentials  of  Algebra,  in  which  the  method  of  presenting  the 
fundamental  topics  differs  at  several  points  from  that  usually  fol- 
lowed.    It  is  simpler  and  more  logical. 

The  latter  chapters  present  such  advanced  topics  as  compound 
interest  and  annuities,  permutations  and  combinations,  continued 
fractions,  summation  of  series,  the  general  theory  of  equations, 
solution  of  higher  equations,  etc. 

Great  care  has  been  taken  to  state  the  various  definitions  and 
rules  with  accuracy,  and  every  principle  has  been  demonstrated 
with  strict  regard  to  the  logical  principles  involved. 

The  examples  and  problems  are  nearly  4,000  in  number,  and 
thoroughly  graded.  They  are  especially  numerous  in  the  impor- 
tant chapters  on  factoring,  fractions  and  radicals. 

The  New  Higher  Algebra  is  adequate  in  scope  and  difficulty  to 
prepare  students  to  meet  the  maximum  requirements  in  ele- 
mentary algebra  for  admission  to  colleges  and  technical  schools. 
The  work  is  also  well  suited  to  the  needs  of  the  entering  classes 
in  many  higher  institutions. 


Half  leather.     Pages,  viii  -\-  4g6.     Introduction  price,  $1.32. 


Wells's  Academic  Algebra.     For  secondary  schools.    $1.08. 

Wells's  Essentials  of  Algebra.     For  secondary  schools.    $1.10. 

Wells's  Higher  Algebra,  $1.32. 

Wells's  University  Algebra.     Octavo.    $1.50. 

Wells's  College  Algebra,  $1.50. 


D.  C.  HEATH  &  CO.,  Publishers,  Boston,  New  York,  Chicago 


COLLEGE    ALGEB 

By  WEBSTER  WELLS,  S.B., 

Professor  of  Mathematics  in  the  Massachusetts  Institute 
of  Technology. 


The  first  eighteen  chapters  have  been  arranged  with  reference 
to  the  needs  of  those  who  wish  to  make  a  review  of  that  portion 
of  Algebra  preceding  Quadratics.  While  complete  as  regards 
the  theoretical  parts  of  the  subject,  only  enough  examples  are 
given  to  furnish  a  rapid  review  in  the  classroom. 

Attention  is  invited  to  the  following  particulars  on  account  of 
which  the  book  may  justly  claim  superior  merit :  — 

The  proofs  of  the  five  fundamental  laws  of  Algebra  —  the  Com- 
mutative and  Associative  Laws  for  Addition  and  Multiphcation,  and 
the  Distributive  Law  for  Multiplication  —  for  positive  or  negative 
integers,  and  positive  or  negative  fractions ;  the  proofs  of  the 
fundamental  laws  of  Algebra  for  irrational  numbers ;  the  proof  of 
the  Binomial  Theorem  for  positive  integral  exponents  and  for 
fractional  and  negative  exponents ;  the  proof  of  Descartes's  Rule 
of  Signs  for  Positive  Roots,  for  incomplete  as  well  as  complete 
equations ;  the  Graphical  Representation  of  Functions  ;  the  so- 
lution of  Cubic  and  Biquadratic  Equations. 

In  Appendix  I  will  be  found  graphical  demonstrations  of  the 
fundamental  laws  of  Algebra  for  pure  imaginary  and  complex 
numbers ;  and  in  Appendix  II,  Cauchy's  proof  that  every  equa- 
tion has  a  root. 

Half  leather.    Pages,  vi  +  ^78.     Introduction  price,  $1. po- 
part I  I,  he  ginning  with  Quadratics.    ^41  pages.    Introduction  price,  $1  .^2. 

D.  C.  HEATH  &  CO.,  Publishers,  Boston,  New  York,  Chicago 


Demonstrations Clear  and  logical 

Problems Well  graded  and  abundant 

Order  of  Topics Suited  to  the  learner 

Scope Adequate  for  the  best  schools 

THESE  are  a  few  of  the  characteristics  of  a  successful  Algebra, 
The  one  book  that  has  them  all,  and  in  the  right  combination  is 
the 

ESSENTIALS  OF  ALGEBRA 

By  Webster  Wells,  S.B. 

Professor  of  Mathematics  in  the  Massachusetts  Institute  of  Technology. 


IT  fully  meets  the  most  rigid  requirements  now  made  in  secondary 
schools.  Like  the  author's  other  Algebras,  it  has  met  with  marked 
success  and  is  in  extensive  use  in  schools  of  the  highest  rank  in  all 
parts  of  the  country. 

The  method  of  presenting  the  fundamental  topics  differs  at  several 
points  from  that  usually  followed.  It  is  simpler,  more  logical  and 
more  philosophical,  yet  by  reason  of  its  admirable  grading  and 
superior  clearness  The  Essentials  of  Algebra  is  not  a  difficult  book. 


Half  leather.    3,$^  pages.     Introduction  Price,  $i.io. 


D.  C.  HEATH  &  CO.,  Publishers,  Boston,  New  York,  Chicago 


Wells's   Mathematical   Series. 

ALGEBRA. 
Wells's  Essentials  of  Algebra      .....     $i.io 

A  new  Algebra  for  secondary  schools.  The  method  of  presenting  the  fundamen- 
tal topics  is  more  logical  than  that  usually  followed.  The  superiority  of  the 
book  also  appears  in  its  definitions,  in  the  demonstrations  and  proofs  of  gen- 
eral laws,  in  the  arrangement  of  topics,  and  in  its  abundance  of  examples. 

Wells's  New  Higher  Algebra       .....       1.32 

The  first  part  of  this  book  is  identical  with  the  author's  Essentials  of  Algebra. 
To  this  there  are  added  chapters  upon  advanced  topics  adequate  in  scope  and 
difficulty  to  meet  the  maximum  requirement  in  elementary  algebra. 

Wells's  Academic  Algebra  .....       1.08 

This  popular  Algebra  contains  an  abundance  of  carefully  selected  problems. 

Wells's  Higher  Algebra    ......       1.32 

»  The  first  half  of  this  book  is  identical  with  the  corresponding  pages  of  the  Aca- 

*%  demic  Algebra.     The  latter  half  treats  more  advanced  topics. 

Wells's  College  Algebra    ......       1.50 

A  modern  text-book  for  colleges  and  scientific  schools.  The  latter  half  of  this 
book,  beginning  with  the  discussion  of  Quadratic  Equations,  is  also  bound  sep- 
arately, and  is  known  as  Wells's  College  Algebra,  Part  II.     $1.32. 

W^ells's  University  Algebra  .....       1.32 

GEOMETRY. 

Wells's  Essentials  of  Geometry  —  Plane,  75  cts.;   Solid,  75  cts.; 

Plane  and  Solid    .......       1.25 

This  new  text  offers  a  practical  combination  of  more  desirable  qualities  than 
any  other  Geometry  ever  published. 

Wells's  Stereoscopic  Views  of  Solid  Geometry  Figures         .         .60 

Ninety-six  cards  in  manila  case. 
Wells's  Elements  of  Geometry  —  Revised  1894.  —  Plane,  75  cts.; 

Solid,  75  cts.;   Plane  and  Solid     .....       1.25 

TRIGONOMETRY. 
Wells's  New  Plane  and  Spherical  Trigonometry  (1896)         .     $1.00 

For  colleges  and  technical  schools.    With  Wells's  New  Six-Place  Tables,  $1.25. 
Wells's  Plane  Trigonometry         .  .  .  .  •         -75 

An  elementary  work  for  secondary  schools.     Contains  Four-Place  Tables. 

Wells's  Essentials  of  Plane  and  Spherical  Trigonometry       .         .go 

For  secondary  schools.  The  chapters  on  Plane  Trigonometry  are  identical  with 
those  of  the  book  described  above.     With  Tables,  $i.o8. 

Wells's  New  Six-Place  Logarithmic  Tables      .  .  .         .60 

The  handsomest  tables  in  print.     Large  Page. 

Wells's  Four-Place  Tables  .  .  .  .  .         .25 

ARITHMETIC. 

Wells's  Academic  Arithmetic      .  .  .  .  .     $1.00 

Correspondence  regarding  ter77is  for  introduction 
and  exchange  is  cordially  invited. 

D.  C.  Heath  &  Co.,  Publishers,  Boston,  New  York,  Chicago 


BOSTON  COLLEGE 


3  9031   01550115  8 


Mathematifsl 


Barton's  Theory  of  Equations.    A  treatise  for  college  classes.    $1.50. 
Bowser's  Academic  Algebra.     For  secondary  schools.    $1.12. 
Bowser's  College  Algebra.     A  full  treatment  of  elementary  and  advanced  topics.     $1.50. 
Bowser's  Plane  and  Solid  Geometry.    $1.25.    Plane,  bound  separately.    75  cts. 
Bowser's  Elements  of  Plane  and  Spherical  Trigonometry.    90  cts.;  with  tables,  $1.40. 
Bowser's  Treatise  on  Plane  and  Spherical  Trigonometry.    An  advanced  work  for  col- 
leges and  technical  schools.     $1.50. 
Bowser's  Five-Place  Logarithmic  Tables.    50  cts. 
Fine'sNumber  System  in  Algebra.    Theoretical  and  historical.    $1.00. 

^.ilbcrfs  Algebra  Lessons.    Thr  ""  "' "1  riimations:    No. 

through  Quadratic  Equations; 
.opkins's  Plane  Geometry, 
'idyland's  Elements  of  the  Co 

■fevre's  Number  and  its  Alg 
i>yman's  Geometry  Exerci^ 
McCurdy's  Exercise  Bool/ 
'?  Plane  and  Spherj 
iih  six-place  tables,  | 
Nichol's  Analytic  Geoip« 
Nichols's  Calculus. 
Osborne's  Differential 

oterson  and  Baldwin 

obbins's  Surveying  a 

chwatt's  Geometric^ 

Waldo's  Descriptive  G 

with  suggestions 

'ells's  Academic  i 

'  ells' s  Essentials 

Veils' s  Academic  l 

vells's  New  Highe 

fells's  Higher  Algftw" 

yells' 8  University  4y 
v/ells's  College  Algel 
Wells's  Essentials  of 
Wells's  Elements  of  ( 
Wells's  New  Plane  ai 
$1.00.    With  six  pi 

v/elis's   Complete  T 

Plane,  bound  saps 

Wells's  New  Six-Pla( 

Wells's  Four-Place  T 

For  Arithn. 


MATH.  DEFT 

c>^^4  Jo  i 


D.C.  HEATH 


BOSTON  COLLEGE  LIBRARY 

UNIVERSITY  HEIGHTS 
CHESTNUT  HILL.  MASS. 

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HEATH'S  MATHEMATICAL  MONOGRAPH! 

ISSUED    UNDER   THE   GENEIL\L   EDITORSHIP   OF 

WEBSTER  WELLS,   S.B. 

Professor  of  Mathematics  in  the  Massachusetts  Institute  of  7'echnology 


It  is  the  purpose  of  this  series  to  make  direct  contrihu- 
tion  to  the  resources  of  teachers  of  mathematics,  by  pre-| 
senting  freshly  written  and  interesting  monographs  upon] 
the  history,  theory,  subject-matter,  and  methods  of  teachq 
ing  both  elementary  and  advanced  topics.  The  first  five] 
numbers  are  as  follows :  — 

1.  FAMOUS   GEOMETRICAL  THEOREMS    AND    PROBLEMS  AND] 

THEIR  HISTORY.     By  William  W.   Rupert,  C.£. 

i.   The  Greek  Geometers,     ii.  The  Pythagorean  Proposition. 

2.  FAMOUS   GEOMETRICAL  THEOREMS.     By  William  W.  Rupert^ 

ii.  The  Pythagorean  Proposition  (concluded),    iii.  Squaring  the  Circle 

8.   FAMOUS  GEOMETRICAL  THEOREMS.     By  William  W.  Rupert.^ 

iv.  Trisection  of  an  Angle,    v.   The  Area  of  a  Triangle  in  Terms  of|1 
its  Sides.  I' 

4.    FAMOUS  GEOMETRICAL  THEOREMS.     By  William  W.  Rltert.  / 

vi.  The  Duplication  of  the  Cube.    vii.    Mathematical  Inscription  upon 
the  Tombstone  of  Ludolph  van  Ceulen. 

6.    ON  TEACHING   GEOMETRY.     By  Florence  Milner. 
Others  in  preparation. 

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D.   C.   HEATH   &   CO.,   Publishers 

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